Improved versions of some Furstenberg type slicing Theorems for self-affine carpets

The uploaded paper (Algom–Wu, arXiv:2107.02068) proves the slicing bounds for Bedford–McMullen carpets with independent exponents, stated as Theorem 1.2, for every non-axis line, with both Hausdorff- and packing-dimension conclusions. The theorem statement appears verbatim in the abstract and in Section 1.1, and the proof is developed via an approximate-square CP-distribution, entropy estimates, and a geometric consequence of Sinai’s factor theorem, assembled in Theorems 3.1 and the Hausdorff analogue (Section 4), yielding the announced inequalities for all u ≠ 0 and all t (not just almost every t) . By contrast, the candidate solution relies on two key claims that are not justified and are generally false in the stated uniformity: (i) it asserts that for any non-axis direction u and for all typical micromeasures ν arising from the carpet sceneries, dim π_u ν = min{1, dim ν}. The paper does not assume such a uniform projection theorem; known projection results in the CP-chain/local-entropy framework are typically for Lebesgue-a.e. direction, not for all non-axis directions. The paper instead uses a different mechanism (Wu’s consequence of Sinai’s factor theorem) precisely to avoid such a uniform projection input . (ii) It then invokes a dimension-conservation principle to pass from average projected dimensions to uniform fiber bounds for all t, again without supplying the needed hypotheses; the paper derives uniform-in-t bounds through an explicit CP-distribution/entropy scheme. The candidate also misattributes a Jensen/concavity step; while the lower bound E[min{1, X}] ≥ E[X]/s_* can be recovered by an extremal argument under 0 ≤ X ≤ s_*, citing Jensen alone gives the wrong inequality direction. These gaps mean the model’s proof is not correct as written, whereas the paper’s argument stands on its own.